23) The event “three heads” would occur in two ways: first, choose a fair coin, then get three heads and one tail tossing the fair coin; and second, choose an unfair coin, then get three heads and one tail tossing the unfair coin:P(3H,T)=P(F)*P(3H,T|F)+P(U)*P(3H,T|U)= (8/10)(4/2^4)+(2/10)(4*0.9^3*0.1) =0.25832

25) First we calculate the probability of having exactly two heads and two tails:P(2H,2T)=P(F)*P(2H,2T|F)+P(U)*P(2H,2T|U)=(8/10)(6/2^4)+(2/10)(4*0.9^2*0.1^2) =0.30648Then, Bayes’ formula immediately gives us the answer:P(F|2H,2T)= P(F)*P(2H,2T|F) / P(2H,2T)=(8/10)(6/2^4)/ 0.30648≈0.6525